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<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.st_eg.tut_mean_intervals"></a><a class="link" href="tut_mean_intervals.html" title="Calculating confidence intervals on the mean with the Students-t distribution">Calculating
          confidence intervals on the mean with the Students-t distribution</a>
</h5></div></div></div>
<p>
            Let's say you have a sample mean, you may wish to know what confidence
            intervals you can place on that mean. Colloquially: "I want an interval
            that I can be P% sure contains the true mean". (On a technical point,
            note that the interval either contains the true mean or it does not:
            the meaning of the confidence level is subtly different from this colloquialism.
            More background information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">NIST
            site</a>).
          </p>
<p>
            The formula for the interval can be expressed as:
          </p>
<div class="blockquote"><blockquote class="blockquote"><p>
              <span class="inlinemediaobject"><img src="../../../../../equations/dist_tutorial4.svg"></span>

            </p></blockquote></div>
<p>
            Where, <span class="emphasis"><em>Y<sub>s</sub></em></span> is the sample mean, <span class="emphasis"><em>s</em></span>
            is the sample standard deviation, <span class="emphasis"><em>N</em></span> is the sample
            size, /α/ is the desired significance level and <span class="emphasis"><em>t<sub>(α/2,N-1)</sub></em></span>
            is the upper critical value of the Students-t distribution with <span class="emphasis"><em>N-1</em></span>
            degrees of freedom.
          </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
              The quantity α is the maximum acceptable risk of falsely rejecting the
              null-hypothesis. The smaller the value of α the greater the strength
              of the test.
            </p>
<p>
              The confidence level of the test is defined as 1 - α, and often expressed
              as a percentage. So for example a significance level of 0.05, is equivalent
              to a 95% confidence level. Refer to <a href="http://www.itl.nist.gov/div898/handbook/prc/section1/prc14.htm" target="_top">"What
              are confidence intervals?"</a> in <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
              e-Handbook of Statistical Methods.</a> for more information.
            </p>
</td></tr>
</table></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
              The usual assumptions of <a href="http://en.wikipedia.org/wiki/Independent_and_identically-distributed_random_variables" target="_top">independent
              and identically distributed (i.i.d.)</a> variables and <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
              distribution</a> of course apply here, as they do in other examples.
            </p></td></tr>
</table></div>
<p>
            From the formula, it should be clear that:
          </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
                The width of the confidence interval decreases as the sample size
                increases.
              </li>
<li class="listitem">
                The width increases as the standard deviation increases.
              </li>
<li class="listitem">
                The width increases as the <span class="emphasis"><em>confidence level increases</em></span>
                (0.5 towards 0.99999 - stronger).
              </li>
<li class="listitem">
                The width increases as the <span class="emphasis"><em>significance level decreases</em></span>
                (0.5 towards 0.00000...01 - stronger).
              </li>
</ul></div>
<p>
            The following example code is taken from the example program <a href="../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>.
          </p>
<p>
            We'll begin by defining a procedure to calculate intervals for various
            confidence levels; the procedure will print these out as a table:
          </p>
<pre class="programlisting"><span class="comment">// Needed includes:</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
<span class="comment">// Bring everything into global namespace for ease of use:</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>

<span class="keyword">void</span> <span class="identifier">confidence_limits_on_mean</span><span class="special">(</span>
   <span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span>           <span class="comment">// Sm = Sample Mean.</span>
   <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">,</span>           <span class="comment">// Sd = Sample Standard Deviation.</span>
   <span class="keyword">unsigned</span> <span class="identifier">Sn</span><span class="special">)</span>         <span class="comment">// Sn = Sample Size.</span>
<span class="special">{</span>
   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>

   <span class="comment">// Print out general info:</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
      <span class="string">"__________________________________\n"</span>
      <span class="string">"2-Sided Confidence Limits For Mean\n"</span>
      <span class="string">"__________________________________\n\n"</span><span class="special">;</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">7</span><span class="special">);</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Number of Observations"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sn</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Mean"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sm</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Standard Deviation"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Sd</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
</pre>
<p>
            We'll define a table of significance/risk levels for which we'll compute
            intervals:
          </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
</pre>
<p>
            Note that these are the complements of the confidence/probability levels:
            0.5, 0.75, 0.9 .. 0.99999).
          </p>
<p>
            Next we'll declare the distribution object we'll need, note that the
            <span class="emphasis"><em>degrees of freedom</em></span> parameter is the sample size
            less one:
          </p>
<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="identifier">Sn</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
            Most of what follows in the program is pretty printing, so let's focus
            on the calculation of the interval. First we need the t-statistic, computed
            using the <span class="emphasis"><em>quantile</em></span> function and our significance
            level. Note that since the significance levels are the complement of
            the probability, we have to wrap the arguments in a call to <span class="emphasis"><em>complement(...)</em></span>:
          </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">T</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
</pre>
<p>
            Note that alpha was divided by two, since we'll be calculating both the
            upper and lower bounds: had we been interested in a single sided interval
            then we would have omitted this step.
          </p>
<p>
            Now to complete the picture, we'll get the (one-sided) width of the interval
            from the t-statistic by multiplying by the standard deviation, and dividing
            by the square root of the sample size:
          </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">T</span> <span class="special">*</span> <span class="identifier">Sd</span> <span class="special">/</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="keyword">double</span><span class="special">(</span><span class="identifier">Sn</span><span class="special">));</span>
</pre>
<p>
            The two-sided interval is then the sample mean plus and minus this width.
          </p>
<p>
            And apart from some more pretty-printing that completes the procedure.
          </p>
<p>
            Let's take a look at some sample output, first using the <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm" target="_top">Heat
            flow data</a> from the NIST site. The data set was collected by Bob
            Zarr of NIST in January, 1990 from a heat flow meter calibration and
            stability analysis. The corresponding dataplot output for this test can
            be found in <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm" target="_top">section
            3.5.2</a> of the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
            e-Handbook of Statistical Methods.</a>.
          </p>
<pre class="programlisting">   __________________________________
   2-Sided Confidence Limits For Mean
   __________________________________

   Number of Observations                  =  195
   Mean                                    =  9.26146
   Standard Deviation                      =  0.02278881


   ___________________________________________________________________
   Confidence       T           Interval          Lower          Upper
    Value (%)     Value          Width            Limit          Limit
   ___________________________________________________________________
       50.000     0.676       1.103e-003        9.26036        9.26256
       75.000     1.154       1.883e-003        9.25958        9.26334
       90.000     1.653       2.697e-003        9.25876        9.26416
       95.000     1.972       3.219e-003        9.25824        9.26468
       99.000     2.601       4.245e-003        9.25721        9.26571
       99.900     3.341       5.453e-003        9.25601        9.26691
       99.990     3.973       6.484e-003        9.25498        9.26794
       99.999     4.537       7.404e-003        9.25406        9.26886
</pre>
<p>
            As you can see the large sample size (195) and small standard deviation
            (0.023) have combined to give very small intervals, indeed we can be
            very confident that the true mean is 9.2.
          </p>
<p>
            For comparison the next example data output is taken from <span class="emphasis"><em>P.K.Hou,
            O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from Statistics
            for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C. Miller and J.
            N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span> The values result
            from the determination of mercury by cold-vapour atomic absorption.
          </p>
<pre class="programlisting">   __________________________________
   2-Sided Confidence Limits For Mean
   __________________________________

   Number of Observations                  =  3
   Mean                                    =  37.8000000
   Standard Deviation                      =  0.9643650


   ___________________________________________________________________
   Confidence       T           Interval          Lower          Upper
    Value (%)     Value          Width            Limit          Limit
   ___________________________________________________________________
       50.000     0.816            0.455       37.34539       38.25461
       75.000     1.604            0.893       36.90717       38.69283
       90.000     2.920            1.626       36.17422       39.42578
       95.000     4.303            2.396       35.40438       40.19562
       99.000     9.925            5.526       32.27408       43.32592
       99.900    31.599           17.594       20.20639       55.39361
       99.990    99.992           55.673      -17.87346       93.47346
       99.999   316.225          176.067     -138.26683      213.86683
</pre>
<p>
            This time the fact that there are only three measurements leads to much
            wider intervals, indeed such large intervals that it's hard to be very
            confident in the location of the mean.
          </p>
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